Math Primer for CG Ref: Interactive Computer Graphics, Chap. 4, E. Angel

Contents Scalar, Vector, Point Change of Basis Frame Change of Frame Affine Sum, Convex Combination, Convex Hull, … Case Study: shooting game

Introduction Three basic data types in CG: scalars, points, and vectors scalar: not a geometric type per se; used in measurement point: a location in space; exist regardless of any coordinate system vector: any quantity with direction and magnitude; does not have fixed location Examine these concepts in a mathematically more rigorous way …

Scalars: Live in Real Space Two fundamental operations are defined between pairs: Addition, multiplication Closure  S,  S,  ·  S  S Commutative   ·  ·  Associative  ·  ·  ·  ·   Distributive  ·  ·  ·  Additive & multiplicative inverse  ·   

Real Analysis The study of real numbers … If you are interested, see Analysis WebNotes:Analysis WebNotes

Vectors: Live in Vector Space Two kinds of entities: Scalar, vector Two operations and corresponding geometric interpretations: v-v addition (head-tail) scalar-v multiplication (scaling of vector) Properties Closure  u,v  V, u+v  V Commutative u+v=v+u Associative  u+(v+w)=(u+v)+w Distributive   (u+v)=  u+  v   )u=  u+  u

Vector Space (cont) Linear combination u=  1 u 1 +  2 u 2 + … +  n u n Linear independent Only set of scalars such that u=  1 u 1 +  2 u 2 + … +  n u n is zero  1 =  2 =…=  n =0 Basis a set of linearly independent vectors that span the space

Vector Space: change of basis represent any vector uniquely in a basis change of basis

Example New Basis

Points: Live in Affine Space Vector space lacks Location, distance, … Concept of coordinate system (frame) Reference point: origin Frame: origin + basis defines position in space Add another entity to vector spaces Point New operations Point – Point  Vector Point + Vector  Point (translation) Note that the following are not defined: point addition multiplication of scalar & point

Affine Space: change of frames Represent point in a frame Change of frame

Euclidean Space Supplement vector space with the notion of distance New operation Inner (dot) product  S  u,v,w  V u · v=v · u (  u+  v) · w=  u · w +  v · w v · v>0 (v  0) 0 · 0=0 u · v=0  u and v are orthogonal |v|=(v · v) ½ u · v=|u||v| cos 

Summary: Mathematical Spaces Real Space (Scalar) Vector Space (Scalar, Vector) Euclidean Space (Scalar, Vector) [distance] Affine Space (Scalar, Vector, Point) [location]

Affine Sum In affine space, point addition is only defined in the following case: Q R P(  ) Point addition is allowed only if their weights add up to one

Affine Sum (cont) More generally, Convex Combination A particular affine sum where weights are non- negative

Convex Hull

Now, how to apply these math Besides change of basis/frame …

Ex: Parametric Equation of a Line Q R S(  ) Q R

Ex: Plane, Triangle, Parallelogram Q R S(  ) P T(  ) (infinite) Plane Parallelogram Triangle

Ex: Homogeneous Coordinates A unified representation scheme in affine space for points and vectors Change of Frame: subsume change of basis

Affine Space: coordinate system fixing the origin of the vectors of coordinate system at P 0 4x1 “ vector ” : Unified vector & point representation homogeneous coordinate distinguish point & vector 4x4 matrix algebra for change in frames Point: [  1  2  3 1] Frame: [u 1 u 2 u 3 P 0 ] Vector: [  1  2  3 0] Frame: [u 1 u 2 u 3 P 0 ]

Ex: Shooting (AABB) Intersection can be checked using 3D Sutherland algorithm

Cohen-Sutherland Line-Clipping Algorithm Trivially accept Trivially reject clipping

Ex: Shooting (OBB) Change of frame then apply Sutherland algorithm in the local frame

Ex: Ray-Triangle Intersection

Ex: Shooting (discrete version)

Ex: Shooting (continuous version)

Exercise Convex Hull: prove convex combination yields the shape you imagine